and \(\hat{\mathcal {D}}(\alpha +\beta )=\hat{\mathcal {D}}(\alpha )\hat{\mathcal {D}}(\beta )e^{-i\text{Im}(\alpha \beta ^{*})}\) . The relation (REF ) essentially implies \(\mathinner {|{\alpha }\rangle }=\hat{\mathcal {D}}(\alpha )\mathinner {|{0}\rangle }\) , where \(\mathinner {|{0}\rangle }\)  is the state, annihilated by the annihilation operator. Therefore, the coherent states can be obtained through the action of the displacement operator on the state \(\mathinner {|{0}\rangle }\)  (see [1]} for more details). As a result, the inner product between two coherent states can be obtained
\(\mathinner {\langle {\beta |\alpha }\rangle } = \mathinner {\langle {0}|}\hat{\mathcal {D}}^{\dagger }(\beta )\hat{\mathcal {D}}(\alpha )\mathinner {|{0}\rangle }=e^{-\frac{|\alpha -\beta |^{2}}{2}+i\text{Im}(\alpha \beta ^{*})}.\) 
